Effective three-body interactions via photon-assisted tunneling in an optical lattice

We present a simple, experimentally realizable method to make coherent three-body interactions dominate the physics of an ultracold lattice gas. Our scheme employs either lattice modulation or laser-induced tunneling to reduce or turn off two-body interactions in a rotating frame, promoting three-body interactions arising from multiorbital physics to leading-order processes. This approach provides a route to strongly correlated phases of lattice gases that are beyond the reach of previously proposed dissipative three-body interactions. In particular, we study the mean-field phase diagram for spinless bosons with three- and two- body interactions and provide a roadmap to dimer states of varying character in one dimension. This toolkit should be immediately applicable in state-of-the-art cold-atom experiments.

Figure 1

Schematic of the laser-assisted tunneling scheme that converts three-body shifts into dominant three-body interactions, in a strongly interacting system, $U\gg J$. (a) By adding a Raman process with frequency ${\omega }_{m1}\sim U$, we do not affect tunneling where an atom in a singly occupied site tunnels into an unoccupied site (or the equivalent process, where an atom from a doubly occupied site tunnels onto a site with a single particle present). (b) However, the sideband gives rise to near-resonant couplings when an atom in a singly occupied site tunnels so as to produce a doubly occupied site. For $U\gg J$ this results in an effective model where the detuning of the sideband provides a new effective two-body interaction $U_{\mathrm{eff}}^{\left(2\right)}=U-{\omega }_{m1}$.

Figure 2

Example parameter values for (a) ${}^{133}$Cs, with scattering length $a_s=350a_0$, where $a_0$ is the Bohr radius, and three-body parameter $L_3=0.5\times {10}^{-25}$ cm${}^6$ s${}^{-1}$, and (b) ${}^{87}$Rb, with scattering length $a_s=100a_0$ and $L_3=2.3\times {10}^{-28}$ cm${}^6$ s${}^{-1}$. Values are shown as a function of the lattice depth $V_0$ in the recoil energy $E_R$ and include the tunneling $J$, unmodified on-site interaction $U$, three-body shift $\delta U_3$, and on-site three-body loss rate ${\gamma }_3$, estimated for atoms in the lowest band of an optical lattice as discussed in the text.

Figure 3

Phase diagram from Gutzwiller mean-field theory for the Bose-Hubbard model with an additional effective three-body repulsion [Eq. (3)], in a homogeneous system. We show the locations of the phase boundaries as a function of $U_{\mathrm{eff}}^{\left(2\right)}/Jz$ and the mean filling factor $n$, and we assume that the probability of occupation for sites with more than three particles is 0. Here we take $U_{\mathrm{eff}}^{\left(3\right)}=50Jz$.

Figure 4

Ground-state correlations in the interacting dimer model [Eq. (4)], for 24 dimers on 48 lattice sites, with box boundary conditions and ${\varepsilon }_i=0$. (a) Example correlation functions $S_{i,j}=\langle d_i^{†}{d}_j\rangle$ (upper lines, indicating DSF order) and $D_{i,j}=\langle d_i^{†}{d}_i{d}_j^{†}{d}_j\rangle -\langle d_i^{†}{d}_i\rangle \langle d_j^{†}{d}_j\rangle$ (lower lines, indicating CDW order) shown for $U_d=J_d$. Thin-dashed lines show linear fits on the double-logarithmic scale. (b) Exponents for algebraic decay of the correlation functions $D_{i,j}$, $K_D$ and $S_{i,j}$, $K_S$. These are determined by fitting to correlation functions starting from site 9 (to avoid boundary effects) and extending over 15 lattice sites. Error bars represent the range of $K$ values fitted by least squares methods to the correlation functions $S_{9,9+l}$ and $D_{9,9+l}$ points over $l=7$–15 sites. These calculations are converged with DMRG bond dimension $\chi =200$.

Figure 5

Adiabatic preparation of dimer states from a Mott insulator in a superlattice. Here we show the results of t-DMRG numerical simulations based on Eq. (3), showing a time-dependent ramp in a superlattice of period 2, from a two-particle Mott insulator in the lowest wells to regimes with different signs of $U_{\mathrm{eff}}^{\left(3\right)}$, which change the strength of the effective off-site interaction between dimers. The superlattice depth is ramped from $V_{\mathrm{SL}}=20J$ to $V_{\mathrm{SL}}=0$ in an exponential ramp with a time constant ${\tau }_{\mathrm{ramp}}=4J^{-1}$. We choose $U_{\mathrm{eff}}^{\left(2\right)}=-8J$ and compute the dynamics for 24 particles on 25 lattice sites with box boundary conditions. (a) Mean on-site occupation $\langle {\widehat{n}}_x\rangle$ at the end of the ramp for occupation $U_{\mathrm{eff}}^{\left(3\right)}>0$ [solid (red) line] and $U_{\mathrm{eff}}^{\left(3\right)}<0$ [dashed (blue) line]. We observe enhanced oscillations characteristic of the stronger repulsive interaction generated when $U_{\mathrm{eff}}^{\left(3\right)}<0$ (see text for details). (b) On-site occupation $\langle {\widehat{n}}_i\rangle$ plotted as a function of time during the ramp for $U_{\mathrm{eff}}^{\left(3\right)}>0$.

Figure 6

Adiabatic preparation of states with CDW and DSF order from a Mott insulator in a superlattice. Here we show the results of t-DMRG numerical simulations based on Eq. (4), showing a time-dependent ramp in a superlattice of period 2, from a Mott insulator with one dimer in each of the lowest wells, for varying $U_d$. The superlattice depth is exponentially ramped from $V_{\mathrm{SL}}=20J_d/2$ to $V_{\mathrm{SL}}=0$ with a time constant ${\tau }_{\mathrm{ramp}}=10J_d^{-1}$, with 24 dimers on 48 lattice sites and box boundary conditions. Correlation functions shown are from the end of a ramp after a time $T=80J_d^{-1}$, with an exponential ramp divided into time steps of time $\delta t=0.1J^{-1}$ and a constant superlattice depth in each step. (a) Example correlation functions $S_{i,j}=\langle d_i^{†}{d}_j\rangle$ (upper lines, indicating DSF order) and $D_{i,j}=\langle d_i^{†}{d}_i{d}_j^{†}{d}_j\rangle -\langle d_i^{†}{d}_i\rangle \langle d_j^{†}{d}_j\rangle$ (lower lines, indicating CDW order) shown for $U_d=J_d$. Thin-dashed lines show linear fits on the double-logarithmic scale. (b) Exponents for algebraic decay of the correlation functions $D_{i,j}$, $K_D$ and $S_{i,j}$, $K_S$. These are determined by fitting to correlation functions starting from site 9 (to avoid boundary effects) and extending over 11 lattice sites. Error bars and numerical parameters are the same as those in Fig. 4.