Population dynamics in a Floquet realization of the Harper-Hofstadter Hamiltonian

We study a Floquet realization of the Harper-Hofstadter model that has recently been implemented in a gas of cold bosonic atoms. We study in detail the scattering processes in this system in the weakly interacting regime due to the interplay of particle interactions and the explicit time dependence of the Floquet states that lead to band transitions and heating. We focus on the experimentally used parameters and explicitly model the transverse confining direction. Based on transition rates computed within the Floquet-Fermi golden rule we obtain band population dynamics which are in agreement with the dynamics observed in experiment. Finally, we discuss whether and how photon-assisted collisions that may be the source of heating and band population dynamics might be suppressed in the experimental setup by appropriate design of the transverse confining potential. The suppression of such processes will become increasingly important as experiments progress into simulating strongly interacting systems in the presence of artificial gauge fields.

Figure 1

Resulting band structure $E_n(k_x,k_y)$ in units of the recoil-energy $E_r={\hslash }^2{k}_r^2/\left(2M\right)$ as a function of momentum $k_x$ and $k_y$ in the reduced BZ of the noninteracting model, Eq. (1), for the parameters $V_x=6E_r$, $V_{xl}=0.8E_r$, $V_y=10E_r$ and $\kappa =0.58\hslash \omega$ and $\hslash \omega =0.72E_r$. The dispersion splits into 4 bands of which the middle two bands are touching at the borders of the reduced BZ and combined into a single superband.

Figure 2

Overlap matrix element $I_{\mathrm{\Delta }m}=|\langle \langle {\mathrm{\Phi }}_{\mathrm{f}}^{\mathrm{\Delta }m}(x,y)|H_{\text{int}}|{\mathrm{\Phi }}_{\mathrm{i}}^0{(x,y)\rangle \rangle |}^2$ of the wave functions of the 2-dimensional model for a transition $(1,1)\to (3,3)$ summed over the final-state momenta and averaged over the initial-state momenta as a function of the photon transfer $\mathrm{\Delta }m$ normalized to the element at $\mathrm{\Delta }m=0$.

Figure 3

Dynamics of the fractional band populations $n_{\mu }={\sum }_{m^z}{N}_{\mu ,m^z}/N_{\text{total}}$ in the Floquet realization of the Harper-Hofstadter model for the experimental parameters with a transverse harmonic confinement summed over the $m^z$ oscillator quantum number. Colors correspond to the bands shown in Fig. 1.

Figure 4

Total absorbed energy $\mathrm{\Delta }E\left(t\right)=E\left(t\right)-E(t=0)$ per particle in units of $E_r$ in the Floquet realization of the Harper-Hofstadter model for the experimental parameters with a transverse harmonic confinement.

Figure 5

(a) Stability of the ground-state band of the Harper-Hofstadter model to different inelastic transitions as indicated in the figure in the case of transverse confinement by an optical lattice $V_{\mathrm{lat}}^z=V_z{cos}^2\left(k_r^{z}z\right)$ of strength $V_z/E_r^z$ with $E_r^z=E_r$. A jump from 0 to 1 indicates the change of allowed to forbidden for the respective transition. (b) The critical confinement strength $V_z^c/E_r^z$ required to suppress the $\mathrm{\Delta }m=\pm 1$ transitions as a function of $E_r^z/E_r$.

Figure 6

Scattering rates ${\mathrm{\Gamma }}_{(n_1,n_2)\to (n_3,n_4)}$ for the band indicated in the figure in the case of transverse confinement by an optical lattice $V_{\mathrm{lat}}^z=V_z{cos}^2\left(k_r^{z}z\right)$ for $E_r^z=E_r$ (a) and $E_r^z=0.36E_r$ (b), corresponding to the minimum of $V_z^c$ observed in Fig. 5.

Figure 7

Stability of the ground-state band of the Harper-Hofstadter model to different inelastic transitions as indicated in the figure in the case of transverse confinement by a harmonic potential $V_{\mathrm{osc}}^z=\frac{1}{2}m{\omega }_{\mathrm{osc}}{z}^2$ (a) and corresponding scattering rates (b), both as a function of the oscillator energy $E_{\mathrm{osc}}=\hslash {\omega }_{\mathrm{osc}}$ in terms of the recoil energy $E_r$.