Quantum magnetism with ultracold bosons carrying orbital angular momentum

We show how strongly correlated ultracold bosonic atoms loaded in specific orbital angular momentum states of arrays of cylindrically symmetric potentials can realize a variety of spin-1/2 models of quantum magnetism. We consider explicitly the dependence of the effective couplings on the geometry of the system and demonstrate that several models of interest related to a general $XYZ$ Heisenberg model with external field can be obtained. Furthermore, we discuss how the relative strength of the effective couplings can be tuned and which phases can be explored by doing so in realistic setups. Finally, we address questions concerning the experimental readout and implementation and we argue that the stability of the system can be enhanced by using ring-shaped trapping potentials.

Figure 1

Quasi-one-dimensional ladder of ring potentials (labeled by the index $j$) obtained by concatenating unit cells (labeled by the index $i$) formed by two rings such that the central angle of the triangles formed by three neighboring rings is $\mathrm{\Theta }$. The origin of the phases is taken along the direction $2i\leftrightarrow 2i+1$ (indicated with blue straight arrows), so that the couplings are real along this direction and all hopping phases appear in the $2i\leftrightarrow 2i-1$ links (indicated with red dashed arrows). The distance between the closest points of the two nearest-neighbor rings is $d$.

Figure 2

Sketch of some of the second-order processes that take place in the Mott-insulator regime and their associated amplitudes. (a) Second-order processes not involving any flipping of the spins. (b) Second-order processes leading to the flipping of one spin at site $2i$. (c) Second-order processes leading to the simultaneous flipping of the spins at sites $2i$ and $2i+1$ (upper figure) and at sites $2i$ and $2i-1$ (lower figure).

Figure 3

Ladder of ring potentials with four sites per unit cell. The origin of phases is taken along the direction $A_i\leftrightarrow B_i\leftrightarrow C_i$ (indicated with blue straight arrows), so that the couplings are taken real along this direction and all hopping phases appear in the $C_i\leftrightarrow D_i\leftrightarrow A_{i+1}$ links (indicated with red dashed arrows). The distance between the closest points of the two nearest-neighbor rings is $d$.

Figure 4

Dependence of the effective couplings of the $XYZ$ model (11) on the inter-ring separation $d$ for (a) rings of $R=2.5\sigma$ and (b) rings of $R=5.0\sigma$. The dashed vertical lines mark the value of $d$ for which the transition of the $XYZ$ model without external field (11) between the $z$-antiferromagnet and the $x$-ferromagnet occurs. (c) Dependence of the ratio $J_1^1/J_3^1$ on the inter-ring separation. The inset shows the dependence of $h^1$ on the inter-ring separation taking $U/J_3^1=20$ for all values of $d$.

Figure 5

Upper plots in (a) and (b): dependence of the energy difference between the ground and first-excited states on the inter-ring distance $d$ for ladders of different sizes formed by rings of radius $R=2.5\sigma$. Lower plots in (a) and (b): correlations between spins 1 and 9 in a ladder of $N=16$ spins with PBC formed by rings of $R=2.5\sigma$. In (a), the central angle of the ladder is $\mathrm{\Theta }=\frac{\pi }{2}$ and, in (b), $\mathrm{\Theta }=0.48\pi$.