Dissipative preparation of fractional Chern insulators

We report on the numerically exact simulation of the dissipative dynamics governed by quantum master equations that feature fractional quantum Hall states as unique steady states. In particular, for the paradigmatic Hofstadter model, we show how Laughlin states can be to good approximation prepared in a dissipative fashion from arbitrary initial states by simply pumping strongly interacting bosons into the lowest Chern band of the corresponding single-particle spectrum. While pure (up to topological degeneracy) steady states are only reached in the low-flux limit or for extended hopping range, we observe a certain robustness regarding the overlap of the steady state with fractional quantum Hall states for experimentally well-controlled flux densities. This may be seen as an encouraging step towards addressing the long-standing challenge of preparing strongly correlated topological phases in quantum simulators.

Figure 1

The band structure for (a) the KM model [35] with $q=2$ and (b) the Hofstadter model [39] with $q=3$. The red arrows indicate the dissipative pumping of particles from higher bands to the lowest band, as described by the Lindblad operators $L_m$ in Eq. (4).

Figure 2

The dynamics Eq. (4) for the KM model. We characterize the dynamics by (a) the overlap Eq. (6) with the exact $\nu =1/2$ Laughlin states and (b) the weight Eq. (7) in the lowest KM band as functions of $\kappa t$. The dotted lines indicate $\mathcal{O}\left(t\right)=1$ and $\mathcal{W}\left(t\right)=1$ in (a) and (b), respectively.

Figure 3

The dynamics Eq. (4) for the Hofstadter model. We characterize the dynamics by (a) the overlap Eq. (6) with the exact $\nu =1/2$ Laughlin states (dashed curves) and the Hofstadter ground states (solid curves) and (b) the weight Eq. (7) in the lowest Hofstadter band as functions of $\kappa t$. The dotted lines indicate $\mathcal{O}\left(t\right)=1$ and $\mathcal{W}\left(t\right)=1$ in (a) and (b), respectively.

Figure 4

The weight $\mathcal{W}$ [Eq. (7)] of the many-body eigenstate of the static Hamiltonian Eq. (1) in the lowest Hofstadter band for $N=3, N_x=2, N_y=3$ at $\varphi =1/3, 1/4$, and $1/5$. Each eigenstate is labeled by its excitation energy $\mathrm{\Delta }E$ with respect to the ground state. The dotted lines indicate the steady state's weight in the lowest Hofstadter band [shown in Fig. 3] for the corresponding system sizes.

Figure 5

The dynamics Eq. (4) for the Hofstadter model with additional NNN hopping. We characterize the dynamics by (a) the overlap Eq. (6) with the $\nu =1/2$ exact Laughlin states (dashed curves) and the ground states of the model (solid curves) and (b) the weight Eq. (7) in the lowest band as functions of $\kappa t$. The discrepancy between the solid curve and the dashed curve is invisible in (a) for $\varphi =1/4$ and $\varphi =1/5$. The dotted lines indicate $\mathcal{O}\left(t\right)=1$ and $\mathcal{W}\left(t\right)=1$ in (a) and (b), respectively.