Spin-orbit coupling in periodically driven optical lattices

We propose a method for the emulation of artificial spin-orbit coupling in a system of ultracold, neutral atoms trapped in a tight-binding lattice. This scheme does not involve near-resonant laser fields, avoiding the heating processes connected to the spontaneous emission of photons. In our case, the necessary spin-dependent tunnel matrix elements are generated by a rapid, spin-dependent, periodic force, which can be described in the framework of an effective, time-averaged Hamiltonian. An additional radio-frequency coupling between the spin states leads to a mixing of the spin bands.

Figure 1

Schematic illustration of the relevant processes, which are captured in the two-component single-particle Hamiltonian. $H_{\mathrm{Kin}}$ represents the spin-independent nearest-neighbor tunneling processes, $H_{\mathrm{rf}}$ describes the radio-frequency coupling of the two spin states, and $H_{\mathrm{Force}}$ corresponds to the spin-dependent driving of the atoms in the lattice. The drive originates from an oscillating magnetic field gradient.

Figure 2

Spin-dependent driving of the atoms with sinusoidal pulses. (a) The amplitude of the magnetic field is modulated around zero with trains of sinusoidal pulses. (b) The resulting SOC strength and (c) the renormalization of the tunneling rate as functions of the forcing parameter $K$. In (b) and (c) the colored lines correspond to different ratios $T_1/T$ of the pulse to hold time.

Figure 3

Dispersion relations of the spin-orbit coupled system for sinusoidal pulses with $T_1/T=0.9$. The color code indicates the respective admixture of the bare spin states [Eq. (20)]. The forcing parameter, Rabi frequency, and detuning are (a) $K=1.5$, $\hslash \Omega =3J$, and $\hslash \Delta =0$; (b) $K=1.5$, $\hslash \Omega =9J$, and $\hslash \Delta =0$; (c) $K=0.9$, $\hslash \Omega =9J$, and $\hslash \Delta =0$; and (d) $K=1.5$, $\hslash \Omega =9J$, and $\hslash \Delta =0.5J$.