Extended Bose-Hubbard models with Rydberg macrodimer dressing

Extended Hubbard models have proven to bear novel phases of matter, but their experimental realization remains challenging. In this work we propose to use bosonic quantum gases dressed with molecular bound states in Rydberg interaction potentials for the observation of these quantum states. We study the molecular Rabi coupling with respect to the effective principal quantum number and trapping frequency of the ground-state atoms for various molecular potentials of rubidium and potassium, and the hereby resulting dressed interaction strength. Additionally, we propose a two-color excitation scheme which significantly increases the dressed interaction and cancels otherwise-limiting ac Stark shifts. We study the various equilibrium phases of the corresponding extended Bose-Hubbard model by means of the cluster Gutzwiller approach and perform time evolution simulations via the Lindblad master equation. We find a supersolid phase by slowly ramping the molecular Rabi coupling of an initially prepared superfluid and discuss the role of dissipation.

Figure 1

(a) Schematic of single-color and two-color coupling scheme for the realization of the macrodimer dressing. While the intermediate-state detuning ${\mathrm{\Delta }}_{\text{1C}}$ is approximately given by half of the energy shift of the potential $U_0$ in the single-color scheme, the two-color scheme benefits from the additional tunability. The potential well is further described by the position of the minimum $R_e$, and its eigenstates are approximately given by the vibrational states with vibrational quantum number $\nu$ and their corresponding vibrational wave function ${\mathrm{\Phi }}_{\nu }\left(R\right)$. The atoms in the electronic ground state are described by the initial relative wave function ${\mathrm{\Phi }}_g\left(R\right)$, which depends on the trapping frequency ${\nu }_{\text{trap}}$. (b) Dressed interaction curve for the potential well Rb1 at $n=36$ and a trapping frequency ${\nu }_{\text{trap}}=130$ kHz. The blue dots represent typical distances in an optical lattice with lattice constant $a_{\text{lat}}=532$ nm. (c) Bosonic atoms trapped in a two-dimensional optical lattice with lattice constant $a_{\text{lat}}$ and tunneling with rate $J$. The dressing results in a distance-specific interaction with tunable spatial profile given by the distance $R_e$ and strength $V$. (d) In this work we focus on three potential wells for ${}^{87}\mathrm{Rb}$ (upper diagram) and ${}^{39}\mathrm{K}$ (lower diagram), which arise from avoided crossing energetically located between the asymptotic pair states $|n{P}_{1/2}n{P}_{1/2}\rangle$, $|n{P}_{1/2}n{P}_{3/2}\rangle$, and $|n{P}_{3/2}n{P}_{3/2}\rangle$ for $n\in [25,75]$

Figure 2

(a), (b) Franck-Condon factor $f_0$ vs effective principal quantum number $n^{*}$ for fixed trapping frequency ${\nu }_{\text{trap}}$ (a) and vs trapping frequency ${\nu }_{\text{trap}}$ for fixed principal quantum number $n=35$ (b). For all macrodimer potentials we find a linear growth of the Franck-Condon factor for increasing effective principal quantum numbers $f_0\propto n^{*}$ which scales with the trapping frequency as $f_0\propto {\nu }_{\mathrm{trap}}^{1/4}$. The gray area denotes the typical range of trapping frequencies in the itinerant regime. (c)–(f) For the following plots we use the potential well Rb1 at $n=35$, a Rabi frequency $\mathrm{\Omega }=2\pi \times 3$ MHz, and the optimum detuning $\delta =\alpha f_0\mathrm{\Omega }$ unless mentioned otherwise. For the studied interaction potential of rubidium, we choose typical values for the electronic coefficient $\alpha =0.5$ [33] and a Rydberg decay rate $\gamma =25{\mathrm{ms}}^{\text{-1}}$ [35]. (c), (d) Single-color ac Stark shift $U_{\text{ac}}^{\text{1C}}(r=0)$ vs $\delta$ (c) and two-color ac Stark shift $U_{\text{ac}}^{\text{2C}}(r=0)$ vs ${\mathrm{\Delta }}_{\text{1C}}$ (d) at the beam center $r=0$. Away from the beam center, the differential light shift induced by the ac Stark shift between neighboring lattice sites [circle markers in inset of (c)] is significant in the single-color scheme. In the two-color scheme, the additional tunability gained through the independent intermediate-state detuning ${\mathrm{\Delta }}_{\text{2C}}$ and coupling strength $\varepsilon \mathrm{\Omega }$ allows us to suppress the total ac Stark effect. (e), (f) The dressing quality (e) for various Franck-Condon factors $f_0$ and the dressed interaction (f) vs coupling strength assuming single-particle and macrodimer losses. The dressing quality is independent of the coupling between the ground and intermediate state, while the interaction profits from smaller detunings.

Figure 3

(a) Equilibrium phase diagram of the extended Bose-Hubbard model for $U=2\pi \times 0.5$ kHz and $\overline{\rho }=0.5$ for both NN (dashed line) and NNN (solid line) interactions. We obtain a density wave (DW) and a supersolid (SS) as well as a homogeneous superfluid (SF) regime. NN interaction results in a checkerboard ordering while NNN interaction leads to striped ordering. (b) Ground-state density pattern of exemplary supersolid states of the phase diagram with additional harmonic potential. The upper figure represents a striped ordered supersolid obtained with NNN interaction, the lower figure a checkerboard ordered supersolid obtained with NN interaction. Here, the average filling $\overline{\rho }=0.5$ refers to the density at the center of the harmonic potential. (c) Phase transition for various fillings with NNN interaction. The solid red line corresponds to the phase boundary obtained through an analytic approach for $\overline{\rho }=1$.

Figure 4

(a) Parameter regime of spontaneous symmetry-broken time evolution ($\mathcal{I}>{\mathcal{I}}_{\mathrm{th}}$) and homogeneous time evolution ($\mathcal{I}<{\mathcal{I}}_{\mathrm{th}}$) given by the critical dressed interaction $V_c$ vs the hopping rate $J$ for different values of the ramping times with ${\mathcal{I}}_{\mathrm{th}}=0.05$. Longer ramping times require larger dressed interaction for the imbalance $\mathcal{I}$ to become finite within the evolution times considered here. (b) Imbalance $\mathcal{I}$ vs time $t$ for an initial SF state for fixed parameters of the extended Bose-Hubbard model $U=2\pi \times 0.5$ kHz, $J=2\pi \times 75$ Hz, $V=2\pi \times 175$ Hz, a bare Rydberg scattering rate ${\mathrm{\Gamma }}_{\mathrm{Ryd}}=4{\mathrm{s}}^{\text{-1}}$, and a molecular scattering rate ${\mathrm{\Gamma }}_{\mathrm{mol}}=8{\mathrm{s}}^{\text{-1}}$ for different values of the ramping times $t_{\text{ramp}}\in [100,400]$ ms. We find a finite imbalance emerging after ramping up the coupling to the macrodimer state. The later onset of a finite imbalance occurs for longer ramping times, and its value decays due to the single-particle and macrodimer losses. (c) Exemplary depiction of the time evolution of (b) with ramping time $t_{\text{ramp}}=200$ ms at times $t\in [0,150,300,450]$ ms, where the occupation number $\rho$ at each site is shown. The average filling $\overline{\rho }=1$ is determined at the center of the trap. Due to the anisotropic harmonic confinement, striped order in $y$ direction becomes more favorable.

Figure 5

Upper diagram depicts Franck-Condon factors $f_{\nu }$ vs vibrational quantum number $\nu$ for the interaction potential Rb1 and fixed $n=55$, trapping frequency ${\nu }_{\text{trap}}=40$ kHz. Lower diagram shows the accumulated contribution of vibrational bound states vs $\nu$ for $\delta =2\pi \times 3$ MHz.

Figure 6

(a)–(d) Position of the potential well $R_e$ (a), shift $U_0$ (b), potential depth $D_e$ (c), and spacing ${\mathrm{\Delta }}_{\nu }$ (d) vs effective principal quantum number $n^{*}$. The scaling relations of $R_e$ and $D_e$ fit a previous study [20]. (e) Franck-Condon factor $f_0$ vs trapping frequency ${\nu }_{\text{trap}}$ for fixed principal quantum number $n=65$.

Figure 7

Single-site-Gutzwiller mean-field calculation of the extended Bose-Hubbard model (left) and cluster Gutzwiller calculations of the SS regime (right) for comparison within the grand-canonical ensemble. Using small clusters incorporates additional quantum fluctuations, which reduce the size of the SS regime. The difference is insignificant between the clusters of size $3\times 2$ and $4\times 2$, and we assume further increasing the cluster size will not influence the phase boundaries.

Figure 8

Time evolution boundaries for different ramping times $t_{\mathrm{ramp}}$ and numerical thresholds ${\mathcal{I}}_{\mathrm{th}}$. A larger threshold (${\mathcal{I}}_{\mathrm{th}}=0.1$) shifts the boundaries previously obtained for a smaller threshold (${\mathcal{I}}_{\mathrm{th}}=0.05$) to higher dressed interactions $V$. The difference between the boundaries does not seem to depend on the hopping rate $J$ but increases with ramping time $t_{\mathrm{ramp}}$.