Tuning Long-Range Fermion-Mediated Interactions in Cold-Atom Quantum Simulators

Engineering long-range interactions in cold-atom quantum simulators can lead to exotic quantum many-body behavior. Fermionic atoms in ultracold atomic mixtures can act as mediators, giving rise to long-range Ruderman-Kittel-Kasuya-Yosida–type interactions characterized by the dimensionality and density of the fermionic gas. Here, we propose several tuning knobs, accessible in current experimental platforms, that allow one to further control the range and shape of the mediated interactions, extending the existing quantum simulation toolbox. In particular, we include an additional optical lattice for the fermionic mediator, as well as anisotropic traps to change its dimensionality in a continuous manner. This allows us to interpolate between power-law and exponential decays, introducing an effective cutoff for the interaction range, as well as to tune the relative interaction strengths at different distances. Finally, we show how our approach allows one to investigate frustrated regimes that were not previously accessible, where symmetry-protected topological phases as well as chiral spin liquids emerge.

Figure 1

(a) Two bosonic atoms (white) separated by distance $r$ and trapped in an optical lattice (red) experience an effective long-range interaction mediated by a Fermi gas trapped in an harmonic potential (blue). (b) The contact Bose-Fermi interactions ($g_{\mathrm{bf}}$) virtually populates the conduction band of the Fermi gas. (c) An additional oscillatory potential induces a gap between the valence and conduction gap, exponentially damping the mediated interactions. (d) By controlling the strength of the trapping potential in an orthogonal direction, ${\omega }_z/{\omega }_x$, one can continuously tune the dimension of the Fermi gas from 1D to 2D, introducing additional modulating frequencies, as will be shown in Fig. 4.

Figure 2

(a) Effective potential between two atoms in one dimension mediated by a Fermi gas trapped in an harmonic trap with (green) and without (red) an extra periodic trap of strength $V_p/V_h=400$. Markers indicate the oscillation maxima. (b) Value of the maxima for increasing values of $V_p/V_h$. The inset shows the decay length $\ell$ of a fitted Yukawa interaction, $\sim \exp [-k_{F}r/(\pi \ell )]/r$. (c–d) Strength of the second ${\upsilon }_2/{\upsilon }_1$ (c) and third ${\upsilon }_3/{\upsilon }_1$ (d) neighbor interactions as a function of $V_p/V_h$ and the effective lattice spacing $k_{F}d$. Here, the bosonic Wannier function $w_{\mathit{j}}(\mathit{r})$ is approximated by a gaussian distribution with, $x_{\text{width}}/d=0.17$, consistent with a lattice wavelength $\lambda =784.7\mathrm{nm}$ for ${}^{87}\mathrm{Rb}$. See Ref. [85] for a more thorough discussion about the experimental parameters chosen for this and the rest of the figures.

Figure 3

(a) Real-space configuration of the ${\mathrm{CDW}}_{\pi }$ (left), where spheres of different sizes correspond to an alternating atomic occupation, and frustrated BOW (right) with a dimerized bond structure. (b) $S(q)$ as a function of $k_{F}d$ for a chain with $L=60$ sites and bosonic density $\rho =1/2$, for $V_p=0$. The phase diagram presents a staircase structure, where every step corresponds to a ${\mathrm{CDW}}_q$ characterized by a peak in $S(q)$. (c) Value of $S$ at the peak, $S_{\max }$ as a function of $k_{F}d$ and $V_p/V_h$, showing how the CDW orders melt for sufficiently large values of $V_p/V_h$, giving rise to a frustrated BOW. (d) $S_{\max }$ and Berry phase $\gamma$ as a function of $k_{F}d$ for $V_p/V_h=680$, showing the nontrivial topological nature of the BOW phase. Real-space configuration of the BOW phase for $L=40$, showing (e) dimerized bonds and (f) localized edge states at the boundaries. Parameters as in Fig. 2.

Figure 4

(a) Mediated potential $F(r)$ as a function of the anisotropy ratio ${\omega }_z/(N{\omega }_x)$ and effective atomic separation $k_{F}r$. Repulsive (attractive) forces are represented in red (blue). (b)–(d) Value of $F(r)$ for three anisotropy ratios (blue) indicated in (a) with black dashed lines, compared to the expected analytical results [Eqs. (5) and (6)] (orange). (e) Cosine transform of $F(r)$ as a function of ${\omega }_z/(N{\omega }_x)$, where the dotted lines correspond to the frequencies $2k_F$ (red) and ${\tilde{k}}_1$ (white). (f), (g) Relation between the nearest-neighbor potentials ${\upsilon }_{1,2,3}$ of a kagome lattice with lattice spacing $d$, as a function of ${\omega }_z/({\omega }_{x}N)$ and $k_{F}d$. Dashed contour follows ${\upsilon }_1=0$. Here, we took $N=250$ and the rest of the parameters as in Fig. 2.