Generation of uniform synthetic magnetic fields by split driving of an optical lattice

We describe a method to generate a synthetic gauge potential for ultracold atoms held in an optical lattice. Our approach uses a time-periodic driving potential based on quickly alternating two Hamiltonians to engineer the appropriate Aharonov-Bohm phases, and permits the simulation of a uniform tunable magnetic field. We explicitly demonstrate that our split-driving scheme reproduces the behavior of a charged quantum particle in a magnetic field over the complete range of field strengths, and obtain the Hofstadter butterfly band structure for the Floquet quasienergies.

Figure 1

(a) Schematic representation of the method. The optical lattice is formed as a two-dimensional standing wave between incoming laser beams, and is periodically accelerated in the $x$ direction, producing an inertial force in the lattice rest frame. Two additional running-wave beams, with a frequency difference $\omega$ and crossing at an angle of $2\theta$, add a spatially dependent phase to the driving. In the Landau gauge the tunneling phases $\chi$ appear only on the horizontal hopping processes (marked with arrows). A linear variation of the phase with $y$, $\chi \left(m\right)=m\Phi$, would produce a uniform flux $\Phi$ threading each plaquette. (b) The split-driving scheme. In the first half-period, an acceleration consisting of a constant and oscillating component is applied in the $x$ direction (upper figure) to induce the Aharonov-Bohm tunneling phases, while the tunneling in the $y$ direction ($J_y$) is suppressed (lower figure). In the second half-period the $x$ coordinate is uniformly accelerated back to its original position, suppressing the $x$ tunneling ($J_x$) via the Wannier-Stark effect, while tunneling in the $y$ direction is restored. This pattern of driving is periodically repeated.

Figure 2

Weak-field results. (a) The system was initialized as a Gaussian wave packet of width ${\sigma }^2=5$, and kicked in the $+y$ direction at $t=0$ by applying a phase imprinting $exp\left(iy\right)$. The center of mass of the wave packet describes a circular orbit, the radius of which is inversely proportional to the effective flux, in analogy to the cyclotron orbit of a charged particle in a uniform magnetic field. (b) Here the system is initialized in the ground state of a parabolic trap potential $V=\kappa r^2/2$, with curvature $\kappa =0.1J$, which is then shifted four lattice spacings to the left. The center of mass traces out a rosette pattern, precisely analogous to the path of a Foucault pendulum subjected to the Coriolis force. Physical parameters: driving frequency $\omega =50J$, driving amplitude $K=0.05\omega$, and $n=1$.

Figure 3

Quasienergy spectrum for high and low frequencies; black (solid) circles denote the bulk states and red (hollow) circles are states with more than 50% of their density localized on the edge. Driving parameters are $n=1$ and $K=0.1\omega$. (a) $\omega =1000J$; for this high frequency, the spectrum is indistinguishable from the “butterfly” spectrum of the true Hofstader-Harper Hamiltonian. (b) $\omega =10J$; some differences appear between the quasienergy spectrum and the Hofstadter butterfly; fewer edge states are visible, and a subset of edge states shows only a weak dependence on the magnetic flux.