FIG. 2: The charge density wave (CDW) phase in a square lattice at n 0 = 1 / 2 with ordering wavevector ⃗ Q n = ( π, π ). (b) valence bond solid (VBS) phases with ordering wavevector ⃗ Q K = ( π, 0) where the kinetic energy 〈 K ij 〉 = 〈 b † $_{i}$b j + h.c. 〉 takes a non-zero constant K in the two sites connected with a dimer, but 0 in the two sites without a dimer. (c) Stripe CDW order at ⃗ Q n = ( π, 0) and (d) Plaquette VBS order at ⃗ Q n = ( π, 0) , (0 , π ). [2, 10–17]. FIG. 1: Optical Bragg scattering of cold atoms moving in 2 dimensional optical lattices. The ⃗ q = ⃗ k 1 − ⃗ k 2 and ω = ω 1 − ω 2 are momentum and energy transfer from the laser beams to the cold atoms respectively. The A stands for a aperture, the PM stands for a Photomultiplier. The off resonant scattering processes lead to the on-site term (a) and the off-site term (b) in Eqn.3. cies ω$_{l}$ and mode functions u$_{l}$ ( ⃗ r ) = e i ⃗ k$_{l}$ · ⃗ r + iφ$_{l}$ . The Rabi frequencies Ω are much weaker than the laser beams ( not shown in Fig.1 ) which form the optical lattices. When it is far off the resonance, the laser light-atom detunings Δ$_{l}$ = ω$_{l}$ − ω$_{a}$ where ω$_{a}$ is the two level energy differ- ence are much larger than the Rabi frequency Ω and the energy transfer ω = ω$_{1}$ − ω$_{2}$ ( See Fig.1a and 1b ), so Δ$_{1}$ ∼ Δ$_{2}$ = Δ. After adiabatically eliminating the up- per level e of the two level atoms, expanding the ground state atom field operator ψ$_{g}$ ( ⃗ r ) = ∑ i b$_{i}$w ( ⃗ r − ⃗ r$_{i}$ ) in Eqn.2 where w ( ⃗ r − ⃗ r$_{i}$ ) is the localized Wannier functions of the lowest Bloch band corresponding to V$_{OL}$ ( ⃗ r ) and b$_{i}$ is the annihilation operator of an atom at the site i in the Eqn.1, then we get the effective interaction between the off-resonant laser beams and the ground level g : neighbor hopping which can be tuned by the depth of the optical lattice potential, the U, V$_{1}$, V$_{2}$ are onsite, nearest neighbor (nn) and next nearest neighbor (nnn) interac- tions respectively, the · · · may include further neighbor interactions and possible ring-exchange interactions. The filling factor n = N$_{a}$/N where N$_{a}$ is the number of atoms and N is the number of lattice sites. The on-site inter- action U can be tuned by the Feshbach resonance [2]. Various kinds of optical lattices such as honeycomb, tri- angular [18], body-centered-cubic [18], Kagome lattices [19] can be realized by suitably choosing the geometry of the laser beams forming the optical lattices. There are many possible ways to generate longer range interac- tion V$_{1}$, V$_{2}$, .... of ultra-cold atoms loaded in optical lat- tices. Being magnetically or electrically polarized, the $^{52}$Cr atoms [20] or polar molecules [21] $^{40}$K + 87 Rb ( or $^{39}$K + 87 Rb ) interact with each other via long-rang anisotropic dipole-dipole interactions. Loading the $^{52}$Cr or the polar molecules on a 2d optical lattice with the dipole moments perpendicular to the trapping plane can be mapped to Eqn.1 with long-range repulsive interac- tions ∼ p $^{2}$/r3 where p is the dipole moment. The CDW supersolid phases studied by QMC [11] and described in [15] by the dual vortex method was numerically found to be stable in large parameter regimes in this system [22]. The generation of the ring exchange interaction has been discussed in [24]. Some of the important phases with long range interactions are listed in Fig.2. Recently, the quantum entanglement properties of the VB state was addressed in [25]. The interaction between the two laser beams in Fig.1 with the two level bosonic atoms is: where Ψ( ⃗ r ) = ( ψ$_{e}$, ψ$_{g}$ ) is the two component boson an- nihilation operator, the incident and scattered lights in Fig.1a and the two incident lights in Fig.1b have frequen- where ⃗ q = ⃗ k$_{1}$ − ⃗ k$_{2}$ , f$_{0}$ ( ⃗ q ) = ∫ d⃗ re − i⃗ q · ⃗ $^{r}$w $^{2}$( ⃗ r ) and n ( ⃗ q ) = 1 N ∑ N i =1 e − i⃗ q · ⃗ $^{r$_{i}$}$n$_{i}$ = ∑ ⃗ k b † ⃗ $_{k}$b$_{⃗}$ k + ⃗ q is the Fourier transform of the density operator at the momentum ⃗ q . Note that n ( ⃗ q ) = n ( ⃗ q + ⃗ K ). The wavevector is confined to L − 1 < q < a − 1 where the trap size L ∼ 100 µm and the lattice constant a ∼ 0 . 5 µm in Fig.1. In fact, more information is encoded in the off-site kinetic coupling in Eqn.3. In a where the interacting matrix element is J$_{i,j}$ = ∫ d⃗ rw ( ⃗ r − r$_{i}$ ) u ∗ $_{1}$( ⃗ r ) u$_{2}$ ( ⃗ r ) w ( ⃗ r − ⃗ r$_{j}$ ) = J$_{j,i}$ . The first term in Eqn.3 is the on-site term ˆ D = ∑ N i J$_{i,i}$n$_{i}$ ( See Fig.1a ). The second term is the off-site term ( See Fig.1b ). Because the Wannier wavefunction w ( ⃗ r ) can be taken as real in the lowest Bloch band, the off-site term can be written as ˆ K = ∑ N <ij> J$_{i,j}$b † $_{i}$b$_{j}$ = ∑ N <ij> J$_{i,j}$ ( b † $_{i}$b$_{j}$ + h.c. ) which is nothing but the off-site coupling to the nearest neighbor kinetic energy of the bosons K$_{ij}$ = b † $_{i}$b$_{j}$ + h.c. . It is easy to show that: H$_{int}$ = $_{ℏ}$Ω 2 Δ e − $^{iωt}$[ N ∑ i J$_{i,i}$n$_{i}$ + N ∑ <ij> J$_{i,j}$b † $_{i}$b$_{j}$ ] (3) ˆ D ( ⃗ q ) = f$_{0}$ ( ⃗ q ) N ∑ i =1 e − i⃗ q · ⃗ $^{r$_{i}$}$n$_{i}$ = N f$_{0}$ ( ⃗ q ) n ( ⃗ q ) (4) H$_{int}$ = ∫ d $^{2}$⃗ r Ψ $^{†}$( ⃗ r )[ ⃗ p 2 2 m$_{a}$ + V$_{OL}$ ( ⃗ r ) + ℏ ω$_{a}$ 2 σ$_{z}$ + Ω 2 ∑ l ( e − iω$_{l}$$^{t}$σ $^{+}$u$_{l}$ ( ⃗ r ) + h.c. )]Ψ( ⃗ r ) (2) 2 q k , 1 1 k , 2 2 A PM a a (a) i j (b) i (a) (b) (c) (d)