Quantum Mechanics with a Momentum-Space Artificial Magnetic Field

The Berry curvature is a geometrical property of an energy band which acts as a momentum space magnetic field in the effective Hamiltonian describing single-particle quantum dynamics. We show how this perspective may be exploited to study systems directly relevant to ultracold gases and photonics. Given the exchanged roles of momentum and position, we demonstrate that the global topology of momentum space is crucially important. We propose an experiment to study the Harper-Hofstadter Hamiltonian with a harmonic trap that will illustrate the advantages of this approach and that will also constitute the first realization of magnetism on a torus.

Figure 1

The numerical energy spacing relative to the lowest eigenvalue, ${\mathcal{E}}_g$, for the lowest 100 states, obtained by diagonalizing Eq. (4) for $\kappa a^2/J=0.02$ and a lattice with $N=41\times 41$ sites. The energy spacing is multiplied by $\alpha$ to highlight the spacing of toroidal Landau levels in the lowest band.

Figure 2

The top (middle) row is the 9th (48th) lowest eigenstate of Eq. (4) for $\alpha =1/11$, $\kappa a^2/J=0.02$, and a lattice of $N=133\times 133$ sites. (a) and (c) The numerical wave function, $|\psi (\mathbf{r})|$, in units of $a^{-1}$. (b) and (d) The numerical population over bands in the MBZ in units of $a$. (b) and (d) are qualitatively indistinguishable from the analytical $\beta =8$ toroidal Landau level, ${\chi }_8(\mathbf{p})$, in the lowest and second lowest Harper-Hofstadter band, respectively. In both bands, $|\mathrm{\Omega }|=a^2/(2\pi \alpha )$ and toroidal Landau levels have the same form (Supplemental Material [41]). (e) Quantitative comparison along $p_x=0$ taken from (b),(d), and the analytical $\beta =8$ level.