Effective time-independent description of optical lattices with periodic driving

Abstract

For a periodically driven quantum system an effective time-independent Hamiltonian is derived with an eigen-energy spectrum, which in the regime of large driving frequencies approximates the quasi-energies of the corresponding Floquet Hamiltonian. The effective Hamiltonian is evaluated for the case of optical lattice models in the tight-binding regime subjected to strong periodic driving. Three scenarios are considered: a periodically shifted one-dimensional (1D) lattice, a twodimensional (2D) square lattice with inversely phased temporal modulation of the well depths of adjacent lattice sites, and a 2D lattice subjected to an array of microscopic rotors commensurate with its plaquette structure. In case of the 1D scenario the rescaling of the tunneling energy, previously considered by Eckardt et al. in Phys. Rev. Lett. 95, 260404 (2005), is reproduced. The 2D lattice with well depth modulation turns out as a generalization of the 1D case. In the 2D case with staggered rotation, the expression previously found in the case of weak driving by Lim et al. in Phys. Rev. Lett. 100, 130402 (2008) is generalized, such that its interpretation in terms of an artificial staggered magnetic field can be extended into the regime of strong driving.

Light-induced gauge fields for ultracold atoms

“…24 (a)], is intimately related to methods exploiting time-periodic modulations ("shaking") of the optical lattice [59,63,96,304,357,367]. This analogy is illustrated in Fig.…”

“…One can generate artificial magnetic flux in such lattices by combining lattices and time-dependent quadrupolar potentials [101,371], by modulating the lattice depth (i.e. the tunneling amplitude) in a directional manner [105] or by shaking optical lattices [59,63,96,304,357,367]. Similarly, spin-orbit couplings could be generated by subjecting an optical lattice to time-dependent magnetic fields [101,203,204].…”

Periodically driven quantum matter: The case of resonant modulations

“…This requires the corresponding HamiltonianĤ (0) to have a unit cell containing only one A and one B site. The wavevector k is then conserved (up to reciprocal lattice vectors) by all the terms in line (30). However, the timemodulation componentsĤ ±1 can have lower spatial symmetry:…”

“…Working to second order in t 01 [Eqn. (30)], the static Hamiltonian leads to dispersions for particles moving on the (decoupled) A and B sublattices of…”

Merging and alignment of Dirac points in a shaken honeycomb optical lattice

“…Because the states with a different index n are separated by an energy which is a multiple of Ω and because the spectrum possesses a Brillouin-zonelike structure, only the terms with n = 0 need to be taken into account. The effective Hamiltonian H eff , which gives rise to the same spectrum as the Floquet Hamiltonian, is then defined by [21] …”