3 matic increase of the scattering cross section across the Mott to the SF transition due to the pref- actor N $^{2}$. This prediction could be tested immediately. Surprisingly, there is no such optical Bragg scattering experiment in the superfluid yet. In the CDW with ⃗ Q$_{n}$ = ( π, π ) in Fig.2a, due to the lack of VBS order on both sides, the second term in Eqn.6 can be neglected, so that square lattice, the bonds are either oriented along the ˆ x axis ⃗ r$_{j}$ − ⃗ r$_{i}$ = ˆ x or along the ˆ y axis ⃗ r$_{j}$ − ⃗ r$_{i}$ = ˆ y , we have: where K$_{α}$ ( ⃗ q ) = 1 N ∑ N i =1 e − i⃗ q · ⃗ $^{r$_{i}$}$K$_{i,i}$$_{+}$$_{α}$ = e iq$_{α}$/ 2 ∑ ⃗ k cos k$_{α}$b † ⃗ $_{k}$b$_{⃗}$ k + ⃗ q are the Fourier transform of the kinetic energy operator K$_{ij}$ = b † $_{i}$b$_{j}$ + h.c. along α = x, y bonds at the momentum ⃗ q and the ”form” factors f$_{α}$ ( ⃗ q ) = f ( ⃗ q, ⃗ r$_{i}$ − ⃗ r$_{j}$ = α ) = ∫ d⃗ re − i⃗ q · ⃗ $^{r}$w ( ⃗ r ) w ( ⃗ r + ⃗ r$_{i}$ − ⃗ r$_{j}$ ). Note that K$_{α}$ ( ⃗ q ) = K$_{α}$ ( ⃗ q + ⃗ K ). Following the harmonic ap- proximation used in [2], we can estimate that f$_{0}$ ( π, 0) ∼ e − 1 $_{4}$( V 0 /E$_{r}$ ) $_{−}$1 / $^{2}$, f$_{x}$ ( π, 0) ∼ ie − 1 $_{4}$( V 0 /E$_{r}$ ) $_{−}$1 / $^{2}$− π 2 4 ( V 0 /E$_{r}$ ) 1 / $^{2}$, so | f$_{x}$ ( π, 0) /f$_{0}$ ( π, 0) | e − π 2 4 √ V 0 /E$_{r}$ where V$_{0}$ and E$_{r}$ = ℏ $^{2}$k$^{2}$/ 2 m are the strength of the optical lattice potential and the recoil energy respectively [2]. The f$_{0}$ ( π, 0) is close to 1 when V$_{0}$/E$_{r}$ > 4. It is instructive to relate this ratio to that of the hopping t over the onsite interaction U in the Eqn.1: | f$_{x}$ ( π, 0) /f$_{0}$ ( π, 0) | ∼ t U a$_{s}$ a where a$_{s}$ is the zero field scattering length and a = λ/ 2 = π/k is the lattice constant, using the typical values t/U ∼ 10 − $^{1}$, a$_{s}$ /a ∼ 10 − $^{2}$, one can estimate | f$_{α}$/f$_{0}$ | ∼ 10 − $^{3}$. Note that the harmonic approximation works well only in a very deep optical lattice V$_{0}$ ≫ E$_{r}$ , so the above value underestimates the ratio, so we expect | f$_{α}$/f$_{0}$ | ≥ 10 − $^{3}$. The differential scattering cross section of the light from the cold atom systems in the Fig.1 can be calcu- lated by using the standard linear response theory: which should show a peak at ⃗ q = ⃗ Q$_{n}$ ( Fig.3a ) whose amplitude scales as the square of the number of atoms inside the trap ∼ | f$_{0}$ ( π, π ) | $^{2}$N $^{2}$m2 where m = n$_{A}$ − n$_{B}$ is the CDW order parameter [15]. When ⃗ q = ⃗ K , then S $_{CDW}$( ⃗ K ) ∼ | f$_{0}$ (2 π, 0) | $^{2}$N $^{2}$n2 where f$_{0}$ (2 π, 0) ∼ f 2 $_{0}$( π, π ) ( Fig.3a ). So the ratio of the two peaks in Fig.3a is ∼ m $^{2}$/n2 if one neglects the very small difference of the two form factors. Slightly away from 1 / 2 filling, the CDW in Fig.2a may turn into the CDW supersolid ( CDW-SS ) phase through a second order phase transition [15]. Then we have 〈 n ( ⃗ q ) 〉 = mδ$_{⃗}$ q, ⃗ Q$_{n}$ + nδ$_{⃗}$ q, 0 where n = n$_{A}$ + n$_{B}$ = 1 / 2 + δn . The superfluid density ρ$_{s}$ ∼ δn = n − 1 / 2. The scattering cross section inside the CDW- SS: S CDW − $_{SS}$( ⃗ Q$_{n}$ ) ∼ | f$_{0}$ ( π, π ) | $^{2}$N $^{2}$m2 stays more or less the same as that inside the CDW, but S CDW − $_{SS}$( ⃗ K ) ∼ | f$_{0}$ (2 π, 0) | $^{2}$N $^{2}$n2 + 2 | f$_{x}$ (2 π, 0) | $^{2}$N $^{2}$( δn ) $^{2}$B2 will increase. The B is the average bond strength due to very small superfluid component ρ$_{s}$ ∼ δn = n − 1 / 2 flowing through the whole lattice. So the right peak in Fig.3a will in- crease due to the increase of the total density and the superfluid component inside the CDW-SS phase. where ⃗ q = ⃗ k$_{1}$ − ⃗ k$_{0}$, ω = ω$_{1}$ − ω$_{2}$ , the S$_{n}$ ( ⃗ q, ω ) = 〈 n ( − ⃗ q, − ω ) n ( ⃗ q, ω ) 〉 is the dynamic density-density re- sponse function whose Lehmann representation was listed in [4]. The S$_{K}$$_{α}$ ( ⃗ q, ω ) = 〈 K$_{α}$ ( − ⃗ q, − ω ) K$_{α}$ ( ⃗ q, ω ) 〉 is the bond-bond response function whose Lehmann rep- resentation can be got from that of the S$_{n}$ ( ⃗ q, ω ) sim- ply by replacing the density operator n ( ⃗ q ) by the bond operator K$_{α}$ ( ⃗ q ). The integrated scattering cross sec- tion over the final energy dσ d Ω = ∫ dE dσ d Ω dE is propor- tional to the equal-time response function dσ d Ω = S ( ⃗ q ) ∼ ( Ω 2 Δ ) $^{2}$N $^{2}$[ | f$_{0}$ ( ⃗ q ) | $^{2}$S$_{n}$ ( ⃗ q ) + ∑ α =ˆ x, ˆ y | f$_{α}$ ( ⃗ q ) | $^{2}$S$_{K}$$_{α}$ ( ⃗ q )]. We first look at the superfluid to Mott transition at We first look at the superfluid to Mott transition at integer filling factor n . When ⃗ q is equal to the short- est reciprocal lattice vector ⃗ K = (2 π, 0), in the Mott state, dσ M d Ω ∼ | f M 0 (2 π, 0) | $^{2}$N $^{2}$n$^{2}$, in the superfluid state, dσ SF d Ω ∼ | f SF 0 (2 π, 0) | $^{2}$N $^{2}$n2 + 2 | f SF x (2 π, 0) | $^{2}$N $^{2}$B2 where B is the average kinetic energy on a bond in the super- fluid side. Because | f SF 0 (2 π, 0) | 2 ∼ | f M 0 (2 π, 0) | 2 ∼ 1 and B is appreciable in the superfluid side, we expect a dra- which should show a peak at ⃗ q = ⃗ Q$_{K}$ signifying the VBS ordering at ⃗ Q$_{K}$ whose amplitude scales also as the square of the number of atoms inside the trap ∼ | f$_{x}$ ( π, 0) | $^{2}$N $^{2}$K2 where K = K$_{x}$ − K$_{y}$ is the VBS or- der parameter [15]. So the ratio of the VBS peak at ⃗ q = ⃗ Q$_{K}$ over the uniform density peak at ⃗ q = ⃗ K is Now we discuss the VBS state with ⃗ Q$_{K}$ = ( π, 0) in Fig.2b. Due to the uniform distribution of the density in the VBS, when ⃗ q = ⃗ K , the second term in Eqn.6 can be neglected, so there is a diffraction peak ( Fig.3b ) whose amplitude scales as the square of the num- ber of atoms inside the trap ∼ | f$_{0}$ (2 π, 0) | $^{2}$N $^{2}$n2 where f$_{0}$ (2 π, 0) ∼ f 4 $_{0}$( π, 0) and n = 1 / 2 is the uniform density in the VBS state. However, when one tunes ⃗ q near ⃗ Q$_{K}$ , the first term in Eqn.6 can be neglected, then dσ d Ω dE | V BS ∼ ( Ω 2 Δ ) $^{2}$N 2 ∑ α =ˆ x, ˆ $_{y}$| f$_{α}$ ( ⃗ q ) | $^{2}$S$_{K}$$_{α}$ ( ⃗ q, ω ) (9) dσ d Ω dE = S ( ⃗ q, ω ) ∼ ( Ω 2 Δ ) $^{2}$N $^{2}$[ | f$_{0}$ ( ⃗ q ) | $^{2}$S$_{n}$ ( ⃗ q, ω ) + ∑ α =ˆ x, ˆ y | f$_{α}$ ( ⃗ q ) | $^{2}$S$_{K}$$_{α}$ ( ⃗ q, ω )] (6) dσ d Ω dE | CDW ∼ ( Ω 2 Δ ) $^{2}$N $^{2}$| f$_{0}$ ( ⃗ q ) | $^{2}$S$_{N}$ ( ⃗ q, ω ) (8) dσ SF d Ω − dσ M d Ω = 2 | f SF x (2 π, 0) | $^{2}$N $^{2}$B2 (7) ˆ K$_{□}$ = N [ f$_{x}$ ( ⃗ q ) K$_{x}$ ( ⃗ q ) + f$_{y}$ ( ⃗ q ) K$_{y}$ ( ⃗ q )] (5)