Dynamics of weakly interacting bosons in optical lattices with flux

Realization of strong synthetic magnetic fields in driven optical lattices has enabled implementation of topological bands in cold-atom setups. A milestone has been reached by a recent measurement of a finite Chern number based on the dynamics of incoherent bosonic atoms. The measurements of the quantum Hall effect in semiconductors are related to the Chern-number measurement in a cold-atom setup; however, the design and complexity of the two types of measurements are quite different. Motivated by these recent developments, we investigate the dynamics of weakly interacting incoherent bosons in a two-dimensional driven optical lattice exposed to an external force, which provides a direct probe of the Chern number. We consider a realistic driving protocol in the regime of high driving frequency and focus on the role of weak repulsive interactions. We find that interactions lead to the redistribution of atoms over topological bands both through the conversion of interaction energy into kinetic energy during the expansion of the atomic cloud and due to an additional heating. Remarkably, we observe that the moderate atomic repulsion facilitates the measurement by flattening the distribution of atoms in the quasimomentum space. Our results also show that weak interactions can suppress the contribution of some higher-order nontopological terms in favor of the topological part of the effective model.

Figure 1

Schematic representation of the model. The unit cells are shaded. (a) Effective Hamiltonian without correction, ${\widehat{H}}_{\mathrm{eff},0}$ (6). Vertical links correspond to real hopping amplitudes (along ${\mathbf{e}}_y$ direction), while the horizontal links to the right of lattice sites labeled $\mathtt{A}$, $\mathtt{B}$, $\mathtt{C}$, and $\mathtt{D}$ correspond to complex hopping amplitudes with phases $\frac{3\pi }{4}$, $\frac{\pi }{4}$, $-\frac{\pi }{4}$, and $-\frac{3\pi }{4}$, respectively (when hopping from left to right). (b) Effective Hamiltonian with correction, ${\widehat{H}}_{\mathrm{eff},1}$ (7). Red lines represent positive next-nearest-neighbor hopping amplitudes (connecting uppercase letters), while the blue lines represent negative next-nearest-neighbor hopping amplitudes (connecting lowercase letters). Nearest-neighbor hopping amplitudes are the same as in (a).

Figure 2

Energy bands of the effective Hamiltonians. (a) ${\widehat{\mathcal{H}}}_{\mathrm{eff},0}\left(\mathbf{k}\right)$ Eq. (B1), which is without the $J_x^2/\omega$ correction term. (b) ${\widehat{\mathcal{H}}}_{\mathrm{eff},1}\left(\mathbf{k}\right)$ Eq. (B2), which includes the correction term. Driving frequency $\omega =20$; gaps are open. (c) Same as (b), but with $\omega =10$. Gaps are closed.

Figure 3

Anomalous drift $x\left(t\right)$. Dashed purple lines: numerical simulations using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$ (cases 2 and 4 from Table 1). Solid green lines: effective Hamiltonians ${\widehat{H}}_{\mathrm{eff},1}$ (c) and (d) and ${\widehat{H}}_{\mathrm{eff},0}$ (a) and (b) (cases 1 and 3). Dotted black lines: theoretical prediction (14) from ${\gamma }_{\mathrm{eff},1}\left(t\right)$ or ${\gamma }_{\mathrm{eff},0}\left(t\right)$. (a) Initial states and band populations obtained using the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff},0}$ without the correction (cases 3 and 4). Driving frequency $\omega =20$. (b) $\omega =10$. (c) Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$ with the $J_x^2/\omega$ correction (cases 1 and 2). Driving frequency $\omega =20$. (d) $\omega =10$.

Figure 4

Time evolution of the filling factor $\gamma \left(t\right)$ for driving frequency $\omega =20$. Solid purple lines: evolution governed by the time-dependent Hamiltonian $\widehat{H}\left(t\right)$ (cases 2 and 4 from Table 1). Dashed green lines: evolution governed by the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$ or ${\widehat{H}}_{\mathrm{eff},0}$ (cases 1 and 3). Dotted black lines: green lines shifted in order to compare them with purple lines. Shift is chosen so that the two lines approximately overlap. (a) Initial states and band populations obtained using the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff},0}$, which is without the $J_x^2/\omega$ correction term (cases 3 and 4). (b) Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$ which is with the correction term (cases 1 and 2).

Figure 5

Effects of interactions. (a) Anomalous drift $x\left(t\right)$ for several different values of the interaction coefficient $U$. $U$ is given in units where $J=1$. Thick lines: numerical simulations using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. Thin lines: theoretical prediction (14) from ${\gamma }_{\mathrm{eff},1}\left(t\right)$. (b) Corresponding ${\gamma }_{\mathrm{eff},1}\left(t\right)={\eta }_1\left(t\right)-{\eta }_2\left(t\right)+{\eta }_3\left(t\right)$, obtained from simulations using the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$.

Figure 6

(a) Evolution of the band populations ${\eta }_i\left(t\right)$. Dashed lines: numerical results obtained using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. Solid black lines: exponential fit using $f\left(t\right)=a+b{e}^{-ct}$. The coefficient $a$ was fixed to $a_1=0.25$, $a_2=0.50$, and $a_3=0.25$ for the first, second, and third band, respectively. (b) Dependence of the exponential decay coefficients for the lowest band population ${\eta }_1\left(t\right)$ on the interaction strength. $U$ is given in units where $J=1$.

Figure 7

Real-space density distribution, noninteracting case $U=0$. (a) Initial state. (b) After $50\mathrm{ms}$ (75 driving periods), evolution using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. (c) Evolution using effective Hamiltonian without correction ${\widehat{H}}_{\mathrm{eff},0}$. (d) Evolution using effective Hamiltonian with correction ${\widehat{H}}_{\mathrm{eff},1}$.

Figure 8

Real-space density distribution after $50\mathrm{ms}$ (75 driving periods), interacting case. $U$ is given in units where $J=1$. (a) Evolution using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$, $U=0.01$. (b) Same with $U=0.05$. (c) Evolution using the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$, $U=0.01$. (d) Same with $U=0.05$.

Figure 9

Atomic cloud width for different interaction strengths, evolution using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. $U$ is given in units where $J=1$. (a) $d_x=\sqrt{\langle x^2\rangle -{\langle x\rangle }^2}$. (b) $d_y=\sqrt{\langle y^2\rangle -{\langle y\rangle }^2}$.

Figure 10

(a) Comparison of anomalous drifts obtained from evolution using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$ (solid purple line), effective Hamiltonian without correction ${\widehat{H}}_{\mathrm{eff},0}$ (dashed green line) and effective Hamiltonian with correction ${\widehat{H}}_{\mathrm{eff},1}$ (dotted black line). Intermediate interaction strength $U=0.01$. $U$ is given in units where $J=1$. (b) Time evolution of the inverse participation ratio in momentum space for several different values of $U$. Evolution is performed using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. When the interactions are strong enough, $\mathrm{IPR}$ approaches the maximal possible value ($10000$ in this case), which is equal to the total number of states and corresponds to the completely delocalized state. $U$ is given in units where $J=1$. (c) Chern number of the lowest band obtained for different interaction strengths as the ratio of the theoretical prediction for the anomalous drift and numerical results: $c_1\left(t\right)=\left(\frac{2F{a}^2}{\pi \hslash }{\int }_0^t{\gamma }_{\mathrm{eff},1}\left(t^{\prime }\right)d{t}^{\prime }\right)/\left[x\left(t\right)-x\left(t_0\right)\right]$.

Figure 11

Lowest band Chern numbers extracted from numerical data for several different values of detuning $\delta$. Purple circles: noninteracting case, $U=0$. Green triangles: $U=0.01$. Blue squares: theoretical values of the lowest band Chern number $c_1^{\prime }$. A topological phase transition is visible at ${\delta }_c\approx 1.38$. The lines between points are only a guide to the eye.

Figure 12

Eight energy subbands of ${\widehat{\mathcal{H}}}_{\mathrm{eff},1}\left(\mathbf{k}\right)$ for the driving frequency $\omega =20$. Subbands 1 and 2 form the lowest band with Chern number $c_1=1$, subbands 3, 4, 5, and 6 form the middle band with $c_2=-2$, and subbands 7 and 8 form the highest band with $c_3=1$.

Figure 13

Population in higher bands, comparison of numerical results (solid line) with the Fermi's golden rule in the first and second approximation (dashed lines). Band populations are calculated for an initial BEC in an eigenstate of the effective Hamiltonian and then averaged over (approximately) all states in the first band. (a) Initial state and evolution from the effective Hamiltonian with correction ${\widehat{H}}_{\mathrm{eff},1}$, Eq. (7). (b) Without the correction, ${\widehat{H}}_{\mathrm{eff},0}$, Eq. (6).

Figure 14

(a) Kinetic energy per particle (expectation value of the time-dependent Hamiltonian $E_{\mathrm{kin}}\left(t\right)=\frac{1}{N}\langle {\sum }_{l,m,i,j}{\psi }_{l,m}^{*}\left(t\right)H_{lm,ij}\left(t\right){\psi }_{i,j}\left(t\right)\rangle$ divided by the total number of particles $N$) for several different interaction strengths. (b) Interaction energy per particle $E_{\mathrm{int}}\left(t\right)=\frac{1}{N}\frac{U}{2}〈{\sum }_{l,m}{\left|{\psi }_{l,m}\left(t\right)\right|}^2\left[|{\psi }_{l,m}{\left(t\right)|}^2-1\right]〉$. $U$ is given in units where $J=1$.

Figure 15

Momentum-space density distribution in all bands, ${\eta }_1\left(\mathbf{k}\right)+{\eta }_2\left(\mathbf{k}\right)+{\eta }_3\left(\mathbf{k}\right)$. $U$ is given in units where $J=1$. Left: evolution using the time-dependent Hamiltonian ${\widehat{H}}_{\mathrm{eff},1}$. Right: evolution using the time-dependent Hamiltonian $\widehat{H}\left(t\right)$. (a), (b) Initial state. (c), (d) Final state after $50\mathrm{ms}$ (75 driving periods), noninteracting case $U=0$. (e), (f) $U=0.01$. (g), (h) $U=0.05$.