Phases of interacting bosons in a hybrid Harper-Hofstadter system with a synthetic dimension of harmonic trap states

Abstract

Synthetic dimensions are a powerful tool for engineering desired quantum systems, based on coupling together sets of states and reinterpreting these as lattice sites along an artificial dimension. Recently, a synthetic dimension of harmonic trap states has been successfully implemented in an ultracold atom experiment, opening the way for future realizations in this platform of topological lattice models, such as hybrid Harper-Hofstadter (HH) systems, which have one real and one synthetic dimension. However, unlike conventional systems, inter-particle interactions along a synthetic dimension of harmonic trap states are inhomogeneous, long-ranged and non-state-preserving. Therefore, this setup provides a natural platform for the exploration of the interplay between long range interactions (including correlated pair tunneling) and magnetic effects. In this paper, we set out to numerically study the effect of such interactions on both a hybrid two-legged HH ladder and a 2D HH model. In the former, we find variants of vortex and Meissner phases familiar from conventional models, while in the latter, we observe the emergence, in small finite systems, of unusual ground states, including a ``Meissner stripe" state, which combines counter-propagating Meissner-like currents with strong density variations. This opens up interesting questions, including about the nature of strongly-correlated states that would emerge in such a platform.

Figures (20)

• Figure 1: A schematic for how to engineer HH physics. (a) Along the $x$ direction, there is a single-band tight-binding lattice, with hopping amplitude $J_x$ and with sites indexed by $n$. (b) Along the $y$ direction, there is a strong harmonic trap with states indexed by $\lambda$. Applying the periodic potential $V(x, y, t)$ [Eq. (\ref{['Eq:realistic_potential']})] leads to an effective Floquet Hamiltonian [Eq. (\ref{['Eq:F']})] in which neighboring trap states are coupled with an amplitude $J_{\lambda}$ and a phase that depends on the position along the $x$ direction. (c) When viewed in the hybrid synthetic-real $\lambda\!-\!x$ plane, this is a Harper-Hofstadter model for a charged particle hopping in the presence of a magnetic flux $\Phi$, controlled by the hopping phase $\varphi$.
• Figure 2: Examples of different types of interactions contained within Eq. (\ref{['Eq:rwaint']}), including i) onsite, ii) + iii) correlated pair tunneling and iv) + v) dipole-like terms. For correlated pair and dipole-like interactions we present two cases for interactions centered around $\lambda = 30$ ii) and iv) and $\lambda=10$ iii) and v). In each case, the upper panel is a schematic with examples of where two atoms could be before and after the interaction. In the lower panel we then plot the corresponding relative interaction strength $U(\lambda_1, \lambda_2, \lambda_3, \lambda_4)$ as a function of either $n$ or $m$ as indicated in the upper panel. Note that no onsite-like terms are included within the correlated pair tunneling and dipole-like interactions to help separate out different effects.
• Figure 3: The chiral current [Eq. (\ref{['eq:chiral']})] (upper panels) and ladder density bias (LDB) [Eq. (\ref{['eq:ldb']})] (lower panels) as a function of magnetic flux $\alpha$ and inter-leg hopping $J_x$, calculated for the imaginary-time-evolved state of a $59$-site ladder with (i) no interactions, (ii) contact, (iii) onsite, (iv) correlated pair tunneling, (v) dipole-like and (vi) full RWA synthetic-dimension interactions, with $g=0.5$ and $t=1$. Silver dots indicate those example states plotted in Figs. \ref{['fig:laddersingleparticle']}-\ref{['fig:ladderrwa']} and in Appendix B. Labels (M), (B) and (V) indicate when the ground-state is in the Meissner, LDB and vortex phase respectively. Note that we have normalized the chiral current to one for presentation purposes. The (upper) dotted line in each panel indicates the critical anisotropy [Eq. (\ref{['eq:critical']})]; this is in good agreement with the maximum saturation of the chiral current as expected Atala_2014. As can be seen, the chiral current does not appear to be significantly affected by the choice of interactions; however, the LDB does change, with a non-zero ladder density bias appearing for contact and on-site synthetic dimension interactions reflecting the presence of the biased ladder phase wei2014theory. In these cases, the lower dotted line is a guide to the eye, indicating the boundary between this and the vortex-lattice phase. Note that the non-zero LDB in the single-particle case (around $J_x\approx 0$ and $\alpha \approx 0.5$) is due to numerical artifacts.
• Figure 4: Example single-particle ground states obtained numerically for $\alpha\!=\!1/3$, $t\!=\!1$ and $N_\lambda=59$ with (i) $J_x\!=\!0.25$, (ii) $J_x\!=\!1$, (iii) $J_x\!=\!2$, (iv) $J_x\!=\!3.75$, i.e. parameters corresponding to the silver dots marked in Fig. \ref{['fig:phasediagram']}(i). The density (current) is represented by the size and color (direction) of the dots (arrows). Note that the local density and current are both normalized for each panel to unity. The upper portion of each panel shows the density for the entire system, while the lower portion is a zoomed-in inset of the density and current for the section highlighted in green. As can be seen, panels (i)-(iii) are in the vortex-lattice phase with circulating plaquettes, while panel (iv) is in the Meissner-phase with a strong chiral current.
• Figure 5: Example ground states obtained numerically for $\alpha\!=\!1/3$, $t\!=\!1$, $N_\lambda \!=\!59$ and standard contact interactions of strength $g\!=\!0.5$, with (i) $J_x\!=\!0.25$, (ii) $J_x\!=\!1$, (iii) $k\!=\!2$, (iv) $J_x\!=\!3.75$, i.e. parameters corresponding to the silver dots marked in Fig. \ref{['fig:phasediagram']}(ii). Most features closely resemble those of the single-particle states in Fig. \ref{['fig:laddersingleparticle']}, except for the appearance of the biased ladder phase in panel (iii) and a broader overall density distribution in all cases due to interactions.