Topological Growing of Laughlin States in Synthetic Gauge Fields

We suggest a scheme for the preparation of highly correlated Laughlin states in the presence of synthetic gauge fields, realizing an analogue of the fractional quantum Hall effect in photonic or atomic systems of interacting bosons. It is based on the idea of growing such states by adding weakly interacting composite fermions along with magnetic flux quanta one by one. The topologically protected Thouless pump (“Laughlin’s argument”) is used to create two localized flux quanta and the resulting hole excitation is subsequently filled by a single boson, which, together with one of the flux quanta, forms a composite fermion. Using our protocol, filling $1/2$ Laughlin states can be grown with particle number $N$ increasing linearly in time and strongly suppressed number fluctuations. To demonstrate the feasibility of our scheme, we consider two-dimensional lattices subject to effective magnetic fields and strong on-site interactions. We present numerical simulations of small lattice systems and also discuss the influence of losses.

Figure 1

(a) The key idea of our scheme is to grow LN states by introducing weakly interacting CFs into the system. This is achieved by adding magnetic flux (arrows) in the center and replenishing the arising hole by a new boson (red bullet). (b) We consider the Hofstadter-Hubbard model (flux $\alpha$ per plaquette). Additional flux $\varphi$ can be introduced in the center by adiabatically changing the complex phase of the hoppings marked with a box. Furthermore, the central site is assumed to be externally accessible for a coherent drive (Rabi frequency $\mathrm{\Omega }$).

Figure 2

Simulation of the full protocol on a ${\mathrm{C}}_{60}$ buckyball as described in the text, for $U=10J$ and including the static potential (5) with $g=J$. The overlaps (solid line, conditioned on the targeted particle number $N$—dotted line) together with particle-number fluctuations (dashed-dotted line) indicate the accuracy of our protocol.