Supersolid phases of ultracold bosons trapped in optical lattices dressed with Rydberg $p$ states

Engineering quantum phases with spontaneously broken symmetries is a major goal of research in different fields. Trapped ultracold Rydberg-excited atoms in optical lattices are a promising platform for realizing quantum phases with broken lattice translational symmetry since they are interacting over distances larger than the lattice constant. Although numerous theoretical works on trapped Rydberg-excited gases have predicted such phases, in particular, density wave or supersolid phases, their experimental observation proves to be difficult due to challenges such as scattering processes and the limited experimentally achievable coupling strength. Most of these previous studies have focused on isotropically interacting gases dressed with Rydberg $s$ states, while the effect of anisotropic interactions due to Rydberg-excited $p$ states in trapped quantum gases remains much less investigated. Additionally, it was shown that the excitation scheme used to excite Rydberg $p$ states possesses advantages regarding achievable coupling strengths and limitation of scattering processes compared to its $s$-state counterpart, which makes the investigation of Rydberg $p$-state dressed quantum gases even more interesting. In the present work we study the extended two-component Bose-Hubbard model, realized with a bosonic quantum gas with Rydberg-excited $p$ states trapped in an optical lattice, within the Gutzwiller mean-field theory. We compute the ground-state phase diagram and investigate its different regimes. By comparison to the phase diagram of the isotropic case, we find the anisotropic interaction to be more advantageous for the observation of supersolid phases.

Figure 1

(a) Orientation of the magnetic field given by the angle ${\theta }_0$. The magnetic field $\mathbf{B}$ serves as a reference axis for the anisotropic interaction. (b) Interaction $V(r,\theta -{\theta }_0)$ for the full range of relative angle $\theta -{\theta }_0$, which is $V_0$ at $\theta -{\theta }_0=n\pi$ and approximately $V_1$ at $\theta -{\theta }_0=(m+1/2)\pi$ with $n,m\in \mathbb{N}$ for $V_0\gg V_1$. (c) Schematic of the processes corresponding to the different terms in the Hamiltonian (1). The particles are subject to on-site potentials given by $\mu$ and $\mathrm{\Delta }$; on-site interactions with strengths $U_g, U_{ge}$, and $U_e$; hopping with strengths $J_g$ and $J_e$; and interstate conversion with Rabi frequency $\mathrm{\Omega }$. (d) Single-photon coupling scheme characterized by the Rabi coupling $\mathrm{\Omega }$, the transition frequency ${\omega }_0$, the laser frequency ${\omega }_l$, and the detuning $\mathrm{\Delta }$.

Figure 2

(a) Exact soft-core potential $U_{\mathrm{ex}}$ of the two-body Hamiltonian (dotted line) and the soft-core potential $U_{\mathrm{ap}}$ within the Hartree- approximation (dashed line) for detuning $\mathrm{\Delta }/\mathrm{\Omega }=1$ and $\theta -{\theta }_0=\pi /2$. (b) Ratio $U_{\mathrm{ap}}/U_{\mathrm{ex}}$ between the exact and the approximated soft-core potentials for various detunings. While for positive detuning both soft-core potentials are in good agreement, we obtain a pronounced difference for small absolute or negative values of the detuning.

Figure 3

The $J_g\text{−}\mu$ phase diagram for the anisotropic long-range interaction at $U_g/\mathrm{\Omega }=0.1, U_{ge}/\mathrm{\Omega }=1, U_e/\mathrm{\Omega }=1000, \mathrm{\Delta }/\mathrm{\Omega }=2, V_0/\mathrm{\Omega }=1000, V_1/\mathrm{\Omega }=1, {\theta }_0=0$, and $J_e/\mathrm{\Omega }=0$. The topology of the phase boundaries resembles the ones of the single-species Bose-Hubbard model, although the vacuum, the Mott insulator, and the superfluid regime have been replaced by two distinct density wave regimes (DW I and DW II) and a supersolid (SS II) regime. The indices denote the superlattice area $A_{\text{SL}}$ and the spatially modulated density is characterized by stripes, parallel to the reference axis in orientation, of excited-state atoms. The distance between stripes does not change here, since the parameters of the Hamiltonian relevant for the Rydberg state are kept fixed.

Figure 4

Mean ground-state order parameter ${\overline{\varphi }}_g$ (dark blue line) and excited-state order parameter ${\overline{\varphi }}_e$ (light red line) at (a) fixed hopping amplitude and (b) fixed chemical potential. The DW-SS phase transition is of second order upon the parameter change, as the mean observables exhibit a kink at the transition.

Figure 5

(a) So-called devil's staircase, (b) mean occupation of the excited state ${\overline{n}}_e$, and (c) excited-state fraction $n_1^e$ of the occupied site of crystalline density wave phases obtained by variation of the detuning $\mathrm{\Delta }$ and various long-range interaction strengths $V_0/\mathrm{\Omega }$ at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, V_1/\mathrm{\Omega }=1, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. While the crystalline structure and mean occupation number depend on the detuning and the long-range interaction strength, we find that the occupation number of the excited state of the lone occupied site within the superlattice unit cell only depends on the detuning.

Figure 6

The $J_g\text{−}\mathrm{\Delta }$ phase diagram for the anisotropic long-range interaction at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, V_0/\mathrm{\Omega }=1000, V_1/\mathrm{\Omega }=1, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. We find a density wave, a supersolid, a vacuum, and a superfluid regime. The SS regime is composed of SS I and SS II phases with different spatial modulations. The subscript indices denote the superlattice area $A_{\text{SL}}$.

Figure 7

Mean ground-state order parameter ${\overline{\varphi }}_g$ (dark blue line) and excited-state order parameter ${\overline{\varphi }}_e$ (light red line) at (a) fixed detuning and (b) fixed hopping amplitude. First-order phase transitions are identified through jumps of the mean observables (solid vertical line) and second-order transitions are identified through kinks (dashed vertical line).

Figure 8

Phase boundaries between the various regimes obtained for anisotropic (solid line), isotropic (dotted line), and tilted (dashed line) anisotropic long-range interactions at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. While the DW-SS phase transition does not shift significantly, we see a more pronounced difference between the three types of long-range interactions regarding the SS-SF phase transition. For a discussion of the full phase diagrams in the isotropic and tilted anisotropic cases, see the Appendix, Sec. 4.

Figure 9

(a) and (b) Soft-core potential $U_{\mathrm{ex}}$ obtained by exact diagonalization of the two-body Hamiltonian (dotted line) and soft-core potential $U_{\mathrm{ap}}$ within the Hartree approximation (dashed line) for different angles $\theta$ at $V_0/\mathrm{\Omega }=1000$. (c) and (d) Ratio $U_{\mathrm{ap}}/U_{\mathrm{ex}}$ between the exact and the approximated soft-core potentials. The approximated soft-core potentials obtained are larger than the ones obtained with the exact Hamiltonian. While the difference is small for positive detunings ($\mathrm{\Delta }/\mathrm{\Omega }=1$), it becomes more important in the regime of negative detunings ($\mathrm{\Delta }/\mathrm{\Omega }=-1/3$). The angle affects the characteristic range $r_c$, but does not influence the quality of the approximation.

Figure 10

Definition of the underlying superlattice of the crystalline structure. The superlattice is defined by two spanning vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. As a measure for identifying quantum phases and differentiating crystalline structures, we determine the superlattice unit cell area $A_{\text{SL}}=|{\mathbf{a}}_1\times {\mathbf{a}}_2|$. In the nontilted anisotropic case the superlattice unit cell area quantifies the distance between Rydberg-excited stripes and in the isotropic case the distance between Rydberg-excited particles. We also define the excited-state fraction $n_1^e$ of the single occupied site in a superlattice unit cell.

Figure 11

Devil's staircase of crystalline density wave phases obtained by variation of the detuning $\mathrm{\Delta }$ at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, V_0/\mathrm{\Omega }=1000, V_1/\mathrm{\Omega }=1, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. Each step corresponds to a density wave phase, which has the lowest energy for the chosen parameters. The insets show the energy $E/\mathrm{\Omega }$ of ${\mathrm{DW}}_{4\left(10\right)}$ I and ${\mathrm{DW}}_{5\left(11\right)}$ I by variation of the detuning $\mathrm{\Delta }$. The energies of both density wave phases cross and the superlattice area changes across the phase transition.

Figure 12

The $J_g\text{−}\mathrm{\Delta }$ phase diagram obtained with the tilted anisotropic long-range interaction (${\theta }_0=\pi /4$) at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, V_0/\mathrm{\Omega }=1000, V_1/\mathrm{\Omega }=1, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. The spatially modulated ground-state phases are characterized by diagonal stripes of Rydberg-excited particles. While the phase diagram resembles the one obtained for the nontilted anisotropic interaction without tilt (see Fig. 6), it lacks a SS I phase. The indices denote the superlattice area $A_{\text{SL}}$.

Figure 13

The $J_g\text{−}\mathrm{\Delta }$ phase diagram obtained with isotropic long-range interaction at $U_g/\mathrm{\Omega }=0.1, U_e/\mathrm{\Omega }=1, U_g/\mathrm{\Omega }=1000, V_0/\mathrm{\Omega }=1, V_1/\mathrm{\Omega }=1000, J_e/\mathrm{\Omega }=0$, and $\mu /\mathrm{\Omega }=-0.25$. The phase diagram exhibits various regimes with different crystalline structures composed of equidistant Rydberg-excited particles. The indices denote the superlattice area $A_{\text{SL}}$. These results are in agreement with the ones obtained in [13].