Collective $P$-Wave Orbital Dynamics of Ultracold Fermions

We consider the nonequilibrium orbital dynamics of spin-polarized ultracold fermions in the first excited band of an optical lattice. A specific lattice depth and filling configuration is designed to allow the $p_x$ and $p_y$ excited orbital degrees of freedom to act as a pseudospin. Starting from the full Hamiltonian for $p$-wave interactions in a periodic potential, we derive an extended Hubbard-type model that describes the anisotropic lattice dynamics of the excited orbitals at low energy. We then show how dispersion engineering can provide a viable route to realizing collective behavior driven by $p$-wave interactions. In particular, Bragg dressing and lattice depth can reduce single-particle dispersion rates, such that a collective many-body gap is opened with only moderate Feshbach enhancement of $p$-wave interactions. Physical insight into the emergent gap-protected collective dynamics is gained by projecting the Hamiltonian into the Dicke manifold, yielding a one-axis twisting model for the orbital pseudospin that can be probed using conventional Ramsey-style interferometry. Experimentally realistic protocols to prepare and measure the many-body dynamics are discussed, including the effects of band relaxation, particle loss, spin-orbit coupling, and doping.

Figure 1

Conceptual schematic. (a) Fermi-Hubbard physics on a single $X\text{−}Y$ plane. The $\Uparrow$ ($X$-excited) and $\Downarrow$ ($Y$-excited) atoms tunnel at rates $J_0$ and $J_1$ along their ground and excited directions, respectively. There is an on-site $p$-wave interaction $U_{\Uparrow \Downarrow }$ between $\Uparrow$, $\Downarrow$ atoms, as well as nearest-neighbor interactions $V_{ee}$, $V_{\Uparrow \Downarrow }$. (b) Bragg dressing coupling $\Uparrow$, $\Downarrow$ can be implemented with beams (shown in green) that copropagate with the lattice beams (red) when the Bragg-laser wavelength is half that of the lattice beams. The out-of-plane lattice beam is not shown. (c) Effective Bloch sphere of the Bragg-dressed spin states. The $\Uparrow$, $\Downarrow$ states are equal superpositions of the two flavors of the dressed basis. Using standard coherent control protocols, any direction of the Bloch vector can be initialized.

Figure 2

(a),(b) Single-particle spectrum $E_{\overrightarrow{k}}^{\pm }={\overline{E}}_{\overrightarrow{k}}\pm \sqrt{{\epsilon }_{\overrightarrow{k}}^2+(\mathrm{\Omega }/2{)}^2}$ and characteristic contrast time evolution for (a) a weak drive $\mathrm{\Omega }/2\lesssim |{\epsilon }_{\overrightarrow{k}}|$ and (b) a strong drive $\mathrm{\Omega }/2\gg |{\epsilon }_{\overrightarrow{k}}|$ for the Fermi-Hubbard+drive model ${\widehat{H}}_{\mathrm{FH}}+{\widehat{H}}_{\mathrm{\Omega }}$ (green) and spin model ${\widehat{H}}_{\mathrm{S}}$ (purple). (c) Benchmark comparison of the two models’ agreement. Both models are evolved from a product state $|{\psi }_0⟩$ with $\theta =0$ to a fixed time $t_f=50/J_1$, and their contrast $C$ is compared with a root-mean-square error $\mathrm{\Delta }C=[(1/t_f){\int }_0^{t_f}dt|(2/L)(C_{\mathrm{S}}-C_{\mathrm{FH}+\mathrm{\Omega }}){|}^2{]}^{1/2}$, truncated to $\min (\mathrm{\Delta }C,0.2)$ for clarity, using a small system $L=3\times 3$. The representative evolutions in panels (a),(b) are indicated by the circle and triangle, respectively. The purple dashed line indicates the collective regime explored further in Fig. 3.

Figure 3

(a) Time evolution of $⟨{\widehat{S}}^x⟩=C\cos (\varphi )$ to measure the density phase shift $\varphi$, comparing the Fermi-Hubbard $\mathrm{model}+\mathrm{drive}$, ${\widehat{H}}_{\mathrm{FH}}+{\widehat{H}}_{\mathrm{\Omega }}$, spin model ${\widehat{H}}_{\mathrm{S}}$, and one-axis twisting model ${\widehat{H}}_{\mathrm{OAT}}$ for system size $L=3\times 3$ and inclination angle $\theta =\pi /4$. The parameters used lie along the purple dashed line in the previous Fig. 2. (b) Time evolution of $⟨{\widehat{S}}^x⟩$ for a larger system of $L=100$, using only the OAT model together with its predicted mean-field behavior.